Removable edges in a 5-connected graph and a construction method of 5-connected graphs

Abstract

An edge e of a k-connected graph G is said to be a removable edge if Ge is still k-connected. A k-connected graph G is said to be a quasi (k + 1)-connected if G has no nontrivial k-separator. The existence of removable edges of 3-connected and 4-connected graphs and some properties of quasi k-connected graphs have been investigated [In this paper, we first investigate the relation between quasi connectivity and removable edges. Based on the relation, the existence of removable edges in k-connected graphs (k 5) is investigated. It is proved that a 5-connected graph has no removable edge if and only if it is isomorphic to K 6 . For a k-connected graph G such that end vertices of any edge of G have at most k − 3 common adjacent vertices, it is also proved that G has a removable edge. Consequently, a recursive construction method of 5-connected graphs is established, that is, any 5-connected graph can be obtained from K 6 by a number of + -operations. We conjecture that, if k is even, a k-connected graph G without removable edge is isomorphic to either K k+1 or the graph H k/2+1 obtained from K k+2 by removing k/2 + 1 disjoint edges, and, if k is odd, G is isomorphic to K k+1 .

Uniformly 3‐connected graphs

“…An important issue in the theory of $$k$$‐connected graphs is to identify a set of graph operations and a set of some simple $$k$$‐connected graphs satisfying some properties $${{\rm{ {\mathcal B} }}}_{k}$$ such that any $$k$$‐connected graph satisfying these properties can be obtained from a graph in $${{\rm{ {\mathcal B} }}}_{k}$$ by repeated applications of the operations. Such structural characterizations have been discovered by Plummer [14] for 2‐connected graphs, Tutte [17] for 3‐connected graphs, Slater [15] for 4‐connected graphs, Su et al [16] and Xu and Guo [18] for $$k$$‐connected graphs, Dirac [6] for minimally 2‐connected graphs, Dawes [5] for minimally 3‐connected graphs, Kriesell [11] 3‐connected triangle‐free graphs, and Göring et al [8] for uniformly 3‐connected graphs.…”

Removable edges of cycles in 5-connected graphs

“…He proved that there always exist removable edges in a 4-connected graph G unless G is a 2-cyclic graph with order 5 or 6, where a 2-cyclic graph is the graph of the square of a cycle [4], and gave a new construction of 4-connected graphs. Then, Xu [10] generalized the concept to k-connected graphs. She proved that there always exist removable edges in a 5-connected graph G unless G is K 6 , and gave a construction of 5-connected graphs.…”